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A341881 Number of ordered factorizations of n into 5 factors > 1.

%I A341881 #13 Feb 16 2025 08:34:01
%S A341881 1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,5,0,
%T A341881 0,0,0,0,0,0,10,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,25,0,0,
%U A341881 0,0,0,0,0,0,0,0,0,10,0,0,0,5,0,0,0,0,0,0,0,20,0,0,0,0,0,0,0,15
%N A341881 Number of ordered factorizations of n into 5 factors > 1.
%H A341881 Alois P. Heinz, <a href="/A341881/b341881.txt">Table of n, a(n) for n = 32..20000</a>
%H A341881 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/OrderedFactorization.html">Ordered Factorization</a>
%F A341881 Dirichlet g.f.: (zeta(s) - 1)^5.
%F A341881 a(n) = -10 * A000005(n) + 10 * A007425(n) - 5 * A007426(n) + A061200(n) + 5 for n > 1.
%p A341881 b:= proc(n) option remember; series(x*(1+add(b(n/d),
%p A341881       d=numtheory[divisors](n) minus {1, n})), x, 6)
%p A341881     end:
%p A341881 a:= n-> coeff(b(n), x, 5):
%p A341881 seq(a(n), n=32..128);  # _Alois P. Heinz_, Feb 22 2021
%t A341881 b[n_] := b[n] = Series[x*(1 + Sum[b[n/d],
%t A341881      {d, Divisors[n] ~Complement~ {1, n}}]), {x, 0, 6}];
%t A341881 a[n_] := Coefficient[b[n], x, 5];
%t A341881 Table[a[n], {n, 32, 128}] (* _Jean-François Alcover_, Feb 28 2022, after _Alois P. Heinz_ *)
%Y A341881 Column k=5 of A251683.
%Y A341881 Cf. A000005, A007425, A007426, A061200, A070824, A074206, A200221, A341880, A341882.
%K A341881 nonn
%O A341881 32,17
%A A341881 _Ilya Gutkovskiy_, Feb 22 2021